Problem 27 - Chemistry and Chemical Reactivity

Question

Consider the isomerization of butane with an equilibrium constant of \(K=2.5 .\) (See Study Question 13.) The system is originally at equilibrium with [butane] \(=\) \(1.0 \mathrm{M}\) and \([\text { isobutane }]=2.5 \mathrm{M}\) (a) If \(0.50 \mathrm{mol} / \mathrm{L}\) of isobutane is suddenly added and the system shifts to a new equilibrium position, what is the equilibrium concentration of each gas? (b) If \(0.50 \mathrm{mol} / \mathrm{L}\) of butane is added to the original equilibrium mixture and the system shifts to a new equilibrium position, what is the equilibrium concentration of each gas?

Step-by-Step Solution

Answer

(a) 1.14 M butane, 2.86 M isobutane; (b) 1.14 M butane, 2.86 M isobutane.

Step 1: Understanding the Equilibrium Constant

The equilibrium constant \(K\) is defined as \(K = \frac{[\text{isobutane}]}{[\text{butane}]}\). Given that \(K = 2.5\), it means that initially the ratio of concentration of isobutane to butane is 2.5.

Step 2: Initial Conditions Analysis

Initially, the concentration of butane is 1.0 M, and the concentration of isobutane is 2.5 M. Confirm that these satisfy the equilibrium condition \(\frac{[2.5]}{[1.0]} = 2.5\), which matches the given equilibrium constant \(K\).

Step 3: Situation After Adding Isobutane

When 0.50 M of isobutane is added, the concentrations become \([\text{butane}] = 1.0\, \text{M}\) and \([\text{isobutane}] = 3.0\, \text{M}\). The system is initially displaced from equilibrium.

Step 4: Setting Up the ICE Table for Added Isobutane

The next move is to find the change in concentrations using an ICE table. Let the change in isobutane be \(-x\) and the change in butane be \(+x\):\[\begin{align*}\text{Initial:} & \quad [\text{butane}] = 1.0, \; [\text{isobutane}] = 3.0 \\text{Change:} & \quad [\text{butane}] = +x, \; [\text{isobutane}] = -x \\text{Equilibrium:} & \quad [\text{butane}] = 1.0 + x, \; [\text{isobutane}] = 3.0 - x\end{align*}\]And the equilibrium condition becomes: \( \frac{3.0 - x}{1.0 + x} = 2.5 \).

Step 5: Solving for the New Equilibrium (Added Isobutane)

Solve the equation \(\frac{3.0 - x}{1.0 + x} = 2.5\):\[3.0 - x = 2.5(1.0 + x)\]\[3.0 - x = 2.5 + 2.5x\]\[0.5 = 3.5x\]\[x = \frac{0.5}{3.5} = \frac{1}{7}\]Thus, the equilibrium concentrations are:\([\text{butane}] = 1.0 + \frac{1}{7} = 1.14\, \text{M}\)\([\text{isobutane}] = 3.0 - \frac{1}{7} = 2.86\, \text{M}\).

Step 6: Situation After Adding Butane

Next, consider the addition of 0.50 M butane, changing concentrations to \([\text{butane}] = 1.5\, \text{M}, [\text{isobutane}] = 2.5 \, \text{M}\). The system is once more displaced from equilibrium.

Step 7: Setting Up ICE Table for Added Butane

Now, set up a new ICE table:\[\begin{align*}\text{Initial:} & \quad [\text{butane}] = 1.5, \; [\text{isobutane}] = 2.5 \\text{Change:} & \quad [\text{butane}] = -y, \; [\text{isobutane}] = +y \\text{Equilibrium:} & \quad [\text{butane}] = 1.5 - y, \; [\text{isobutane}] = 2.5 + y\end{align*}\]Equilibrium condition is \( \frac{2.5 + y}{1.5 - y} = 2.5 \).

Step 8: Solving for the New Equilibrium (Added Butane)

Solve \( \frac{2.5 + y}{1.5 - y} = 2.5 \):\[2.5 + y = 2.5(1.5 - y)\]\[2.5 + y = 3.75 - 2.5y\]\[3.5y = 1.25\]\[y = \frac{1.25}{3.5} = \frac{5}{14}\]Thus, the new equilibrium concentrations are:\([\text{butane}] = 1.5 - \frac{5}{14} = 1.14\, \text{M}\)\([\text{isobutane}] = 2.5 + \frac{5}{14} = 2.86\, \text{M}\).

Key Concepts

Equilibrium Constant

The equilibrium constant, denoted as \( K \), is a crucial factor in understanding chemical reactions at equilibrium. It is a numeric value representing the ratio of product concentration to reactant concentration at a state of equilibrium. For the isomerization of butane to isobutane, the equilibrium constant is given by:
\[K = \frac{[\text{isobutane}]}{[\text{butane}]}\]
Here, \(K = 2.5\) suggests that at equilibrium, the concentration of isobutane is 2.5 times that of butane. This constant is specific to the reaction and depends only on the nature of reactants and products, temperature, and pressure. Such constants do not change with concentration changes but guide how the system reacts to disturbances, tending to maintain the established ratio at equilibrium.

Concentration Change

Concentration change plays a vital role in how an equilibrium system responds to external adjustments. When concentrations of one or more involved species are altered, the system no longer remains in equilibrium and acts to restore it.
For instance, if more isobutane is added, its concentration increases temporarily. This increase shifts the concentration ratio willingly away from the equilibrium constant, nudging the system to adjust by converting some isobutane back into butane, thus restoring the balance. The opposite occurs if butane is added; isobutane is favored until the equilibrium constant ratio is once again satisfied. Understanding how concentration shifts impact equilibrium helps predict the direction owing to the change.

ICE Table Analysis

The ICE table is an analytical tool used to predict concentration changes in a reaction at equilibrium. It stands for Initial, Change, and Equilibrium, breaking down each state of concentration visually. This method helps simplify the calculations needed to restore equilibrium after a disruption.

• Initial: Represents initial concentrations before any change.
• Change: Denotes the shift in concentration as movement begins toward a new equilibrium state, often expressed in terms of \(x\) or \(y\).
• Equilibrium: Shows the resulting concentrations once equilibrium is re-established.

The utility of the ICE table means that even when concentrations are adjusted, one can solve algebraic equations effectively to find the new equilibrium concentrations. This calculation reinforces understanding of dynamic balance in chemical systems.

Le Chatelier's Principle

Le Chatelier's Principle is an essential concept for explaining how equilibrium systems react to changes. It states that if a system at equilibrium is subjected to a change, the system will adjust itself to counteract that change and restore a new equilibrium.
Additional isobutane shifts its concentration upwards, so the system reacts to lessen the imbalance by converting some isobutane back into butane. Similarly, adding butane tips the equilibrium in the other direction, promoting the formation of more isobutane until equilibrium is achieved. This principle allows us to predict the qualitative effects of concentration, pressure, or temperature transformations. It elucidates how equilibrium can adapt quickly and flexibly to maintain the critical balance dictated by the equilibrium constant.